Solar-interplanetary modeling: 3-D solar wind solutions in prescribed non-radial magnetic field geometries

A model is presented which describes the 3-dimensional non-radial solar wind expansion between the Sun and the Earth in a specified magnetic field configuration subject to synoptically observed plasma properties at the coronal base. In this paper, the field is taken to be potential in the inner corona based upon the Mt. Wilson magnetograph observations and radial beyond a certain chosen surface. For plasma boundary conditions at the Sun, we use deconvoluted density profiles obtained from synopticK-coronameter brightness observations. The temperature is taken to be 2 × 10⁶ K at the base of closed field lines and 1.6 x 10⁶K at the base of open field lines. For a sample calculation, we employ data taken during the period of the 12 November 1966 eclipse. Although qualitative agreement with observations at 1 AU is obtained, important discrepancies emerge which are not apparent from spherically symmetric models or those models which do not incorporate actual observations in the lower corona. These discrepancies appear to be due to two primary difficulties - the rapid geometric divergence of the open field lines in the inner corona as well as the breakdown in the validity of the Spitzer heat conduction formula even closer to the Sun than predicted by radial flow models. These two effects combine to produce conductively dominated solutions and lower velocities, densities, and field strengths at the Earth than those observed. The traditional difficulty in solar wind theory in that unrealistically small densities must be assumed at the coronal base in order to obtain observed densities at 1 AU is more than compensated for here by the rapid divergence of field lines in the inner corona. For these base conditions, the value ofβ(ratio of gas pressure to magnetic pressure) is shown to be significantly greater than one over most of the lower corona - suggesting that, for the coronal boundary conditions used here, the use of a potential or force-free magnetic field configuration may not be justified. The calculations of this paper point to the directions where future research on solar-interplanetary modelling should receive priority:

1. better models for the coronal magnetic field structure
2. improved understanding of the thermal conductivity relevant for the solar wind plasma.

     

Coronal streamers

This work extends a previous analysis of helmet streamers into the somewhat higher range of coronal temperature where streamer geometries are shown to be open, in the sense that there is solar wind expansion everywhere. It is shown that, for a given photospheric field distribution, a certain minimum temperature is required for this type of streamer - this minimum temperature coinciding with the maximum temperature compatible with a helmet streamer. Near this minimum temperature, the streamer is very constricted and the critical point in the streamer core lies at the point of minimum cross-section. Hence the throat, under these conditions, becomes a true geometrical throat rather than the conventional gravitational throat. As the temperature is increased, the streamer shape becomes correspondingly more radial and the location of the throat becomes asymptotically more gravitationally determined. Residual manifestations of coronal streamers at large distances are investigated. It is found that lateral density variations at the earth's orbit tend to be small but velocity variations can become appreciable (100–200 km/sec) for streamers originating in regions where the photospheric magnetic field is strong. At large distances, either streamer or interstreamer regions can dominate, the former occurring at high temperature (2 × 10⁶K) and the latter being favored at lower temperature (1.5 × 10⁶K). In all cases the cross-section becomes essentially radial just above the point where it is a minimum. The marked sensitivity of these shapes to coronal temperature is pointed out - computations indicating that streamers can vary from helmet configurations to almost radial filaments for a very slight increase in temperature. This behavior suggests a strong solar cycle influence upon coronal form.     

On neutral sheets in the solar wind

The structure and dynamics of neutral sheets in the solar wind is examined. The internal magnetic topology of the sheet is argued to be that of thin magnetic tongues greatly distended outward by the expansion inside the sheet. Due to finite conductivity effects, outward flow takes place across field lines but is retarded relative to the ambient solar wind by the reverse J×B force. The sheet thickness as well as the internal transverse magnetic field are found to be proportional to the electrical conductivity to the inverse one third power. Estimating a conductivity appropriate for a current carried largely by the ions perpendicular to the magnetic field, we find sheet dimensions of the order of 500km representative for the inner solar corona. For a radial field of strength 1/2G at 2R , the transverse field there is about 2 × 10^–3G and decreases outward rapidly.The energy release in the form of Joulean dissipation inside the sheet is estimated. It is concluded that ohmic heating in current sheets is not a significant source of energy for the overall solar wind expansion, mainly because these structures occupy only a small percentage of the total coronal volume. However, the local energy release through this mechanism is found to be large - in fact, over 7 times that expected to be supplied by thermal conduction. Therefore, ohmic heating is probably a dominant energy source for the dynamical conditions within the sheet itself.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Evolution of rising helical prominences in a nonuniform atmosphere

Recent coronagraph observations of rising priminences such as in the 14 April and 5 May, 1980 coronal transient events, as well as other older observations, have shown evidence for helical structure in the prominences. If this is true, then a study of the dynamical evolution of rising helical structures in a nonuniform atmosphere is worthwhile. For this study, three important considerations become apparent: (1) Since the ends of the prominence remain rooted in the photosphere, significant stretching of the configuration will result as it rises, (2) due to the fall-off with height of the external quantities, such as gas and magnetic pressure, the prominence will experience time-varying boundary conditions as it rises, and (3) significant lateral expansion of the prominence is expected as the external conditions weaken with height. The interplay of all these effects togehter result in a quite complex dynamical behavior of the prominence.We have tried to obtain some insight into this general problem through a simple model - that of a helical pinch rising in a low beta atmosphere under the influence of an ambient external magnetic field which declines in strength with radial distance from the solar center. Under the general assumptions of an internal uniform, but time-varying, temperature and neglecting gravitational stratification within the prominence, expressions are derived for associated variations in the prominence structure as it rises. We discuss in some detail, particular quantities which are potentially most accessible to observation such as prominence radius, density, and pitch angle as they vary with height during the eruptive process.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

On the reality of potential magnetic fields in the solar corona

Global magnetic field calculations, using potential field theory, are performed for Carrington rotations 1601–1610 during the Skylab period. The purpose of these computations is to quantitatively test the spatial correspondence between calculated open and closed field distributions in the solar corona with observed brightness structures. The two types of observed structures chosen for this study are coronal holes representing open geometries and theK-coronal brightness distribution which presumably outlines the closed field regions in the corona. The magnetic field calculations were made using the Adams-Pneuman fixed-mesh potential field code based upon line-of-sight photospheric field data from the KPNO 40-channel magnetograph. Coronal hole data is obtained from AS&E's soft X-ray experiment and NRL's Heii observations and theK-coronal brightness distributions are from HAO'sK-coronameter experiment at Mauna Loa, Hawaii.The comparison between computed open field line locations and coronal holes shows a generally good correspondence in spatial location on the Sun. However, the areas occupied by the open field seem to be somewhat smaller than the corresponding areas of X-ray holes. Possible explanations for this discrepancy are discussed. It is noted that the locations of open field lines and coronal holes coincide with the locations ofmaximum field strength in the higher corona with the closed regions consisting of relatively weaker fields.The general correspondence between bright regions in theK-corona and computed closed field regions is also good with the computed neutral lines lying at the top of the closed loops following the same general warped path around the Sun as the maxima in the brightness. One curious feature emerging from this comparison is that the neutral lines at a given longitude tend systematically to lie somewhat closer to the poles than the brightness maxima for all rotations considered. This discrepancy in latitude increases as the poles are approached. Three possible explanations for this tendency are given: perspective effects in theK -coronal observations, MHD effects due electric currents not accounted for in the analysis, and reported photospheric field strengths near the poles which are too low. To test this latter hypothesis, we artificially increased the line-of-sight photospheric field strengths above 70° latitude as an input to the magnetic field calculations. We found that, as the polar fields were increased, the discrepancy correspondingly decreased. The best agreement between neutral line locations and brightness maxima is obtained for a polar field of about 30 G.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A new technique for the determination of coronal magnetic fields: A fixed mesh solution to Laplace's equation using line-of-sight boundary conditions

A new method for computing potential magnetic field configurations in the solar atmosphere is described. A discrete approximation to Laplace's equation is solved in the domain R r R ₁, 0 , 0 2 (R ₁being an arbitrary radial distance from the solar center). The method utilizes the measured line-of-sight magnetic fields directly as the boundary condition at the solar surface and constrains the field to become radial at the outer boundary, R ₁. First the differential equation and boundary conditions are reduced to a set of two-dimensional equations in r, by Fourier transforming out the periodic dependence. Next each transformed boundary condition is converted to a Dirichlet surface condition. Then each two-dimensional equation with standard Dirichlet-Dirichlet boundary conditions is solved for the Fourier coefficient it determines. Finally, the solution of the original three dimensional equation is obtained through inverse Fourier transformation. The primary numerical tools in this technique are the use of a finite fast Fourier transform technique and also a generalized cyclic reduction algorithm developed at NCAR. Any extraneous monopole component present in the data can be removed if so desired.The code was developed for the HAO solar-interplanetary modeling effort in response to the following specific requirements:

Some general properties of helmeted coronal structures

A simplified analysis of helmeted coronal structures is carried out and some of the gross properties of such structures discussed. It is found that the magnetically closed region can have but a limited extension into the corona. For temperatures in excess of 1.5 × 10⁶ °K, the maximum height above the limb is about 1.6 R . The maximum possible extension of the helmet from the solar center is exactly one-half the distance to the critical point (where the flow velocity passes through the speed of sound). For this reason, a helmet streamer, at least out to a few solar radii, is essentially a magnetostatic structure - the flow adjacent to the helmet having little effect upon its properties. For given base dimensions, there is a maximum temperature for which a helmet streamer can exist - giving an indication of why such streamers do not appear over young active regions. If the temperature in the helmet and in the streaming region are approximately the same, the helmet height, helmet shape, external flow velocity, and rate of outward decline in the magnetic field are shown to be much more dependent upon the photospheric field distribution than upon the field strength. The density enhancement, however, is a strong function of the field strength. This enhancement is preserved out to the top of the helmet with both the density inside and outside decreasing approximately as predicted by hydrostatic equilibrium. The possible existence of both domed helmets and cusped helmets is demonstrated with the former existing at lower temperatures and the latter at higher temperatures. Cusped helmets occur, however, over a relatively narrow temperature range and are, hence, expected to be less common. The expansion velocity outside the helmet is higher than that predicted by radial flow but increases outward much more slowly. The magnetic field decreases outward proportionally to the square root of the density and inversely proportionally to the velocity - bearing, in general, no relation to a potential field since the rate of decline in field strength is determined by the temperature.On leave from AC Electronics Research Laboratories Santa Barvara, Calif., U.S.A.     

The solar wind and the temperature-density structure of the solar corona

The influence of the solar wind on large-scale temperature and density distributions in the lower corona is studied. This influence is most profoundly felt through its effect upon the geometry of coronal magnetic fields since the presence of expansion divides the corona into magnetically open and closed regions. Each of these regions is governed by entirely different energy transport processes. This results in significant temperature differences since only the open field regions suffer outward conductive heat losses. Because the temperature influences the density in an exponential manner, large density inhomogeneities are to be expected.An approximate method for calculating the temperature and density distribution in a known magnetic field geometry is outlined and numerical estimates are carried out for representative coronal conditions. These estimates show that temperature differences of a factor of about two and density differences of ten can be expected in the lower corona even for uniform base conditions. As a result, we do not regard the so-called coronal holes necessairly as locations of reduced mechanical heating. Alternatively, we suggest that they are regions of open magnetic field lines being continuously drained of energy contert by the solar wind expansion and outward thermal conduction.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Eruptive prominences and coronal transients

The observed interrelationship between coronal transients and eruptive prominences is used as the basis for a theoretical MHD model of these events. The model begins with an equilibrium configuration consisting of a coronal loop or arcade with a filament lying underneath with its axis oriented perpendicular to the overlying field. The lifting of the filament from the solar surface produces an increase in magnetic pressure under the helmet which drives it outward. This increased pressure is associated with the internal field of the filament as well as the field beneath it. The underlying field could be that which produced the filament eruption or, alternatively, reconnected field lines formed by the inward collapse of the legs of the transient towards the neutral line beneath the rising prominence. We do not attempt to explain the filament eruption which may be due to internal forces in the prominence or, alternatively, from forces imposed from beneath as would be produced by emerging flux. In the latter case, the filament is passive and merely acts as a tracer for the more fundamental underlying process.It is shown that the outward force per unit mass produced by the driving magnetic field and the inward restoring forces in the overlying field due to magnetic tension and gravity all decrease with distance at the same rate - namely, as the inverse square of the distance from the solar center. Hence, the ratio of net outward to inward force is independent of radial distance from the Sun. A stability analysis shows that this situation is one of neutral stability.A mathematical model of this physical process is described in which the MHD equations in simplified form, neglecting gas pressure forces, are solved in time for the velocity, width, density, and magnetic field strength of the transient. The solutions show that the velocity increases sharply close to the Sun but quickly approaches a constant value. The width increases linearly with radial distance. Both of these results are in agreement with observations. An examination of the forces exerted on the legs of the transient shows that their motion should be horizontally inward.On leave from the High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colo., U.S.A.     

Particle motions in coronal streamers and type III radio bursts

Both individual and collective motions of electron and proton streams in the current sheet which is thought to exist near the center of a coronal streamer are considered. Unlike previous analyses, closed field lines which must exist when finite conductivity is taken into account as well as a B _ø field due to solar rotation are present. It is shown on the basis of individual particle motions that neither electrons nor protons could move in most of the sheet in the manner required to explain type III bursts since they are effectively tied to the closed field lines.The possibility that the stream could collectively drag the closed field lines out with itself is considered. It is shown that impossibly high densities are required for electron streams and improbable densities for proton streams. Thus the particles responsible for type III bursts cannot travel in the densest part of a coronal streamer, but presumably travel close to this region. Moreover, the current sheet cannot act as a channeling agent to help explain the transverse coherency of type III burst sources.The National Center for Atmospheric Research is sponsored by the National Science Foundation.